Matter-wave interferometry is a powerful tool for high-precision measurements of the quantum properties of atoms, many-body phenomena and gravity. The most precise matter-wave interferometers exploit the excellent localization in momentum space and coherence of the degenerate gases. Further enhancement of sensitivity and reduction of complexity are crucial conditions for success and widening of their applications. Here we introduce a multi-path interferometric scheme that offers advances in both these aspects. Coherent coupling between Bose-Einstein condensates in different Zeeman states is used to generate high-harmonic output signals with an enhanced resolution and maximum possible interferometric visibility. We demonstrate a realisation of such an interferometer as a compact, easy-to-use, atom-chip device. The interferometric scheme is then applied to the reading of the relative phase of two laser pulses.